Computability and Beltrami fields in Euclidean space

TL;DR

This paper demonstrates that certain steady solutions of the Euler equations in Euclidean space can simulate universal Turing machines, revealing deep connections between fluid dynamics and computational complexity, with implications for the space-bounded Church-Turing thesis.

Contribution

It constructs Beltrami flows in Euclidean space that are Turing complete and can simulate arbitrary Turing machines, advancing the understanding of computational phenomena in fluid dynamics.

Findings

Existence of Turing complete steady Euler flows in Euclidean space.

Simulation of arbitrary tape-bounded Turing machines by Beltrami flows on the torus.

High computational complexity in Gaussian random Beltrami fields with probability 1.

Abstract

In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of $\mathbb R^3$, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on $\mathbb T^3$ (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on $\mathbb T^3$ to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability $1$. Finally, our proof also yields Turing complete flows and diffeomorphisms on $\mathbb{S}^2$ with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.